Citation 数理解析研究所講究録 (1999), 1078:

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1 ON COMPLEX MANIFOLDS POLARIZED BY A TitleLINE BUNDLE OF SECTIONAL GENUS $q(x resolution of defining ideals of pr Author(s) Fukuma, Yoshiaki Citation 数理解析研究所講究録 (1999), 1078: Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 ON COMPLEX MANIFOLDS POLARIZED BY AN AMPLE LINE BUNDLE OF SECTIONAL GENUS $q(x)+2$ YOSHIAKI FUKUMA ( ) Naruto University of Education $\mathrm{e}$-mail: fukuma@naruto-u.ac.jp Let $X$ be a smooth projective variety defined over the complex number field, and let be a line bundle over $X$. Then (X, ) is called a polarized (resp. quasi-polarized) manifold if is ample (resp. nef and big). For such pair (X, ), the delta genus $\triangle(l)$ and the sectional genus $g(l)$ are defined by the following formula: $\triangle(l):=n+l^{n}-h^{0}(l)$, $g(l):=1+ \frac{1}{2}(k\mathrm{x}+(n-1)l)ln-1$, where $h^{0}(l)=\dim H^{0}(L)$, and $K_{X}$ is the canonical divisor of $X$. In this report, we will state some recent results about the sectional genus of quasi-polarized manifolds, and we will propose some conjectures and problems. The following results are known for the fundamental properties of the sectional genus; (1) The value of $g(l)$ is a non-negative integer. (Fujita [Fj1], $)$ Ionescu [I] (2) There exists a classification of polarized manifolds (X, ) with the sectional genus $g(l)\leq 2$. (Fujita [Fjl], [Fj2], Ionescu [I], Beltrametti-Lanteri-Palleschi [BLP], e.t.c.)

3 94 (3) Let (X, ) be a polarized manifold with $\dim X=n$. For any fixed and $g(l)$, there are only finitely many deformation types of polarized masnifolds except scrolls over $n$ smooth curves. (For the definition of deformation types of polarized manifolds, see Chapter II, \S 13 in [Fj4].) Here we give the definition of a scroll over a variety. Definition. Let (X, ) be a quasi-polarized manifold with $\dim X=$ $N$, and let $Y$ be a projective variety with $\dim Y=m$ and $N>$ $m\geq 1$. Then (X, ) is called a scroll over if there exists a $Y$ $\pi$ surjective morphism : $Xarrow Y$ such that any fiber $F$ $\pi$ of is isomorphic to and $\mathrm{p}^{n-m}$ $L_{F}\cong O(1)$. Here we consider (3) above. By (3), if (X, ) is not a scroll over a smooth curve, then the topological invariant of $X$ is expected to be bounded by using some invariant of. Here we mainly consider the irregularity $q(x):=\dim H^{1}(Ox)$ of $X$. If (X, ) is a scroll over a smooth curve, then we can easily get that $g(l)=q(x)$. So by considering the fact (3) above, Fujita propose the following conjecture; $([\mathrm{f}\mathrm{j}3])$ Conjecture 1. Let (X, ) be a quasi-polarized manifold. Then $g(l)\geq q(x)$. For the time being, this conjecture is true if (X, ) is one of the following; (1) $n=2,$, $\kappa(x)\leq 1([\mathrm{F}\mathrm{k}2])$ (2) $n=2,$ $\kappa(x)=2,$, $h^{0}(l)\geq 1([\mathrm{F}\mathrm{k}2])$ (3) $n=3,$, $h^{0}(l)\geq 2([\mathrm{F}\mathrm{k}7])$ (4) $\kappa(x)=0,1,$, $L^{n}\geq 2([\mathrm{F}\mathrm{k}3])$ (5) $\dim$ Bs $ L \leq 1$ (For the case in which $\dim$ Bs $ L \leq 0$, see [Fk6], and for the case in which $\dim$ Bs $ L =1$, this result is unpublished). So our first goal is to prove that this conjecture is true if $n=2$, $\kappa(x)=2$, and $h^{0}(l)=0$. Here we give some comments about Conjecture 1. First, for a $(\mathrm{x},\mathrm{l})$ polarized surface with we explain a relation between the value of $g(l)-q(x)$ and the type of divisor $D\in L $. $\kappa(x)\geq 0$ Let

4 \mathrm{b}\mathrm{l}\mathrm{e}$ reduced 95 (X, ) be a quasi-polarized surface with $\kappa(x)\geq 0$ and $h^{0}(l)>0$. Let $D$ be an effective divisor on $X$ which is linearly equivalent to $C_{i}$. Let $D= \sum_{i}a_{i}c_{i}$, where is an curve and $\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{i}\prime We take a birationa 1 morphism $\mu$ $X^{\alpha}arrow X$ $i$ $a_{i}>0$ for each... : such that $C_{1}^{*}\cap C_{2}^{*}\cap C_{3}^{*}=\phi$ for any distinct three irreducible components and of, and if two irreducible curves $\mu^{*}(d)$ $C_{1}^{*},$ $C_{2}^{*}$ $C_{3}^{*}$ $C_{i}^{*}$ $C_{j}^{*}\cdot \mathrm{o}\mathrm{f}\mu^{*}(d)$ and intersect at, then the intersection number $x$ $i(c_{i}^{*}, c_{j}*;x)=1$, where is the intersection number of $i(c_{i}^{*}, C_{j}^{*} ; x)$ $C_{i}^{*}$ $C_{j}^{*}$ and at $x\in C_{i}^{*}\cap C_{j}^{*}$. Let $\mu_{i}$ : $X_{i}arrow X_{i-1}$ be one point $D_{\mathrm{r}\mathrm{e}\mathrm{d}}=B_{0}$ blowing up such that $\mu=\mu_{1}0\mu_{2}0\cdots\circ\mu_{t}$ and let. Let and $(\mu_{i}^{*}(b_{i-1}))_{\mathrm{r}\mathrm{e}\mathrm{d}}=b_{i}$ $B_{i}=\mu_{i}^{*}(B_{i-1})-b_{i}E_{i}$ $E_{i}$, where is a $(- 1)$ -curve such that $\mu_{i}(e_{i})=$ point. Then $b_{i}\geq 1$. Let $X^{\gamma}arrow X^{\beta}$ $\mu^{\gamma}$ $D^{\beta} = \sum_{i}c_{\beta,i}$ and : be a $\mathrm{r}..\mathrm{e}$solution of singular points $S= \bigcup_{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}(\dot{c}_{\beta,i})$. $X_{x}^{\beta,i,t_{x}}arrow X_{x}^{\beta,i,t_{x}1}-arrow\cdotsarrow X_{x}^{\beta,i,0}$ Let $\mu_{x}$ : be a resolution of $\mu_{x}^{k}$ $x\in \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}(c\beta,i)$ $X_{x}^{\beta,i,k}arrow X_{x}^{\beta,i,k-1}$ singularity at, where : is one point blowing up. Let $(\mu_{x}^{k})^{*}(d\beta,k-1)=d^{\beta,k}+m(k, X)Ek$, where is $E^{k}$ $(- 1)$-curve of $\mu_{x}^{k}$ $\mu_{x}^{k}(e^{k})=\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}$ such that is the strict transform of a.n$\mathrm{d}d^{\beta,k}$ $D^{\beta,k-1}$ for each. $k$ By using the above notation, we get the following result; $([\mathrm{f}\mathrm{k}5])$ Theorem 1. Let (X, ) be a polarized surface. Assume that $\kappa(x)\geq 0$ and $h^{0}(l)>0$. Let $D\in L $ be an effective divisor which is linearly equivalent to L. holdsj Then the following inequality $g(d) \geq q(x)+\sum_{\in xjsk}\sum_{=1}\frac{m(k,x_{j})(m(k,x_{j})-1)}{2}+t_{x}j\sum_{1i=}^{t}\frac{b_{i}(b_{i}-1)}{2}$. Proof. See [Fk5]. By this theorem, if the value of $g(l)-q(x)$ is small, then the $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}^{d}$ singularities of is simple.

5 $\overline{l}$ $\overline{l}$ $\overline{l}$ $\overline{l}$ 96 Next we consider the dimension of the global section of the adjoint bundle $K_{X}+(n-1)L$. The value of $g(l)-q(x)$ is thought to control the value of $h^{0}(k_{x}+(n-1)l)$. In [Fk6], the author proposed the following conjecture; $([\mathrm{f}\mathrm{k}6])$ Conjecture 2. Let (X, ) be a quasi-polarized manifold with $\dim X=n$. Then the follwoing inequality $holds_{f}$ $h^{0}(k_{x}+(n-1)l)\geq g(l)-q(x)$. For the time being, this conjecture is true if (X, ) the following; is one of (1) $\dim$ Bs $ L \leq 0$, (2) $\dim X=2$. If Conjecture 1 is true, then the following natural problem arises; Problem 1. For small non-negative integer $m$, give a classification of quasi-polarized manifolds (X, ) with $m=g(l)-q(x)$. If $m=0$ and $n=2$, then this problem relates to the problem of blowing up of polarized surfaces. Let be a smooth projective $S$ surface and let be an ample line bundle on $X$. Let $p_{1},$ $\ldots$, $p_{r}$ be points on in a general position, and let $S$ $\overline{s}arrow S$ $\pi$ : be blowing ups at $p_{1},$ $\ldots$, $p_{r}$. Let $a_{1\cdots)}a_{r}$ be positive integers and $:=$ $\pi^{*}l-\sum_{j}a_{j}ej$, where. $E_{j}:=\pi^{-1}(p_{j})$ Then it is difficult to check that is ample. For the case where $a_{1}=\cdots=a_{r}=1$, Yokoyama proved the following theorem; Theorem 2. (Yokoyama) Assume that $a_{1}=\cdots=a_{r}=1$ $ L $ has an irreducible reduced curve. If $g(l)>q(x)$, then ample. and is When we use this theorem, it is important to know the classification of polarized surfaces $(S, L)$ with $g(l)=q(s)$. Remark. If $\kappa(x)\geq 0$ and has a singularity, then (see [Fkl] and [Fk2]). an irreducible reduced curve $C\in L $ is ample because $g(l)>q(s)$ in this case

6 97 Here we consider the classification of quasi-polarized manifolds (X, ) with small value $m=g(l)-q(x)$. First we study the case in which $X$ is a surface. Then the following facts are known; (2-0-1) A classification of quasi-polarized surfaces (X, ) with $\kappa(x)\leq$ $1$ and $g(l)=q(x)([\mathrm{f}\mathrm{k}2])$. (2-0-2) A classification of quasi-poalrized surfaces (X, ) with $\kappa(x)=$ $2,$ $h^{0}(l)\geq 1$, and $g(l)=q(x)$ ( $[\mathrm{f}\mathrm{k}1]$, [Fk8]). (2-1) A classification of polarized surfaces (X, ) with $\kappa(x)\geq 0$, $h^{0}(l)\geq 1$, and $g(l)=q(x)+1([\mathrm{f}\mathrm{k}5])$. Next we consider the case in which $\dim X=3$. (3-0) A classification of polarized 3-folds (X, ) with $h^{0}(l)\geq 3$ and $g(l)=q(x)([\mathrm{f}\mathrm{k}7])$. In this case (X, ) is one of the following two types; $(3-0_{-}1)$ Polarized 3-folds (X, ) with $\triangle(l)=0$. (This was classified by Fujita. See [Fj4].) (3-0-2) A scroll over a smooth curve. (3-1) A classification of polarized 3-folds (X, ) with $h^{0}(l)\geq 4$ and $g(l)=q(x)+1([\mathrm{f}\mathrm{k}4])$. 3-fold. Then (X, ) a Del Pezzo By considering (3-0) and (3-1), in [Fk4] and [Fk7] the author proposed the following conjecture; Conjecture 3. ([Fk4], [Fk7].) Let (X, ) with $n=\dim X\geq 4$. be a polarized manifold (n-o) Assume that $g(l)=q(x)$ and $h^{0}(l)\geq n$. Then (X, ) is a polarized manifold with $\triangle(l)=0$ or a scroll over a smooth curve. (n-1) Assume that $g(l)=q(x)+1$ and $h^{0}(l)\geq n+1$. Then (X, ) is a $Del$ Pezzo manifold. By considering (3-0) and (3-1) above, we expect that we can classify polarized 3-folds (X, ) with $g(l)=q(x)+2$ and $h^{0}(l)\geq$ 5. The following result is one of the main theorems of the author s talk.

7 . 98 Main Theorem 1. Let (X, ) be a polarized 3-fold. Assume that $h^{0}(l)\geq 5$ and $g(l) q(x)+2$. Then (X, ) is one of the following; (1) A hyperquadric fibration over $\mathrm{p}^{1}$ (2) A scroll over a smooth surface $S$ with $q(s)=0$. Remark. In each cases, the irregularity of $X$ is zero. Hence we get $g(l)=2$. Therefore we obtain an explicit classification of (X, ). (See [Fj2].) Proof of Main Theorem 1. Here we get a sketch of proof of the $\mathrm{n}\mathrm{e}\mathrm{f}$ Main Theorem 1. First assume that $K_{X}+2L$ is not (X, ) is one of the following types: (1) $(\mathrm{p}^{3}, O(1))$, (2) $(\mathrm{q}^{3}, O(1))$, (3) scroll over a smooth curve. Then But in these cases, we obtain $g(l)=q(x)$ and this is a contradiction by hypothesis. $\mathrm{n}\mathrm{e}\mathrm{f}$ So we may assume that $K_{X}+2L$ is. Let $(X, L )$ be the first reduction of (X, ). (Let $X$ be a smooth projective variety with $\dim X=n$ and let be an ample line bundle on $X$. Then we call that $(X, L )$ is the first reduction of (X, ) if there exist a smooth projective variety $X $, an ample line bundle $L $ on $X $, $\pi$ and a birational morphism : $Xarrow X $ $\pi$ such that is a blowing up at a finite set on $X,$ $K_{X}+(n-1)L=\pi^{*}(KX +(n-1)l )$, and $K_{X }+(n-1)l $ is ample.) We remark that $L^{n}\leq(L )^{n}$ in this case. Here we use the following Theorem, which is very important for the proof of Main Theorem. Theorem A. Let (X, ) be a polarized 3-fold with $g(l)=q(x)+$ $m,$ $h^{0}(l)\geq m+3$, and $q(x)\geq m-1$, where $m$ is a non-negative integer. Assume that $K_{X}+L$ is $nef$. Then $L^{3}\leq 2m$. By using Theorem A and the theory of $\triangle$-genus, we can prove the following Claim.

8 , 99 Claim B. $K_{X }+L $ is not $nef$. Proof of Claim $B$. $\mathrm{n}\mathrm{e}\mathrm{f}.$ Assume that $K_{X }+L $ is. $\mathrm{a}$ If $q(x)\geq 1$, then by Theorem we get that $L^{3}\leq(L )^{3}\leq 4$. If $q(x)=0$, then $L^{3}\leq(L )^{3}\leq 2$ $\mathrm{n}\mathrm{e}\mathrm{f}$ since $K_{X }+L $ is. We set $t=4-l^{3}$. Then $t=0,1,2$ or 3. Since $h^{0}(l)\geq 5$, we get $\triangle(l)=3+l3-h0(l)$ $=7-t-h^{0}(L)$ $\leq 2-t$. If $t>0$, then. By using the theory of $\triangle$-genus, we can $\triangle(l)\leq 1$ easily get a contradiction. So we assume $t=0$. If $h^{0}(l)\geq 6$, then we get and by $\triangle(l)\leq 1$ using the same method as above, we get a contradiction. If $h^{0}(l)=5$, then. Here we also use the $\triangle$-genus theory, $\triangle(l)\leq 2$ we also get a contradiction. $\mathrm{n}\mathrm{e}\mathrm{f}$ Therefore $K_{X }+L $ is not. By adjunction theory, polarized manifolds (X, ) such that $K_{X }+L $ is not nef is classified. (1) $K_{X}\sim-2L$, that is, (X, ) is a Del Pezzo manifold. (2) A hyperquadric fibration over a smooth curve. (3) A scroll over a smooth surface. (4) Let $(X, L )$ be the first reduction of (X, ). (4-1), $(X, L/)=(\mathrm{Q}^{3}, \mathcal{o}(2))$ (4-2) $(X, L )=(\mathrm{p}^{3}, O(3))$, (4-3) $X $ is a $\mathrm{p}^{2}$-bundle over a smooth curve $C$ with $(F,$ $L _{F^{\prime)}}=$ $(\mathrm{p}^{2}, \mathcal{o}(2))$ for any fiber $F $ of it. In the end we check these cases in detail, and we obtain the result. Next we consider the case where $\dim X\geq 3$. In particular, we mainly consider the case in which Bs $ L =\emptyset$. Then we get the following results; Let (X, ) be a polarized manifold such that Bs. $ L =\emptyset$ (f-o) If $g(l)=q(x)$, then $\triangle(l)=0$ or (X, ) is a scroll over a smooth curve. (f-1) If $g(l)=q(x)+1$, then (X, ) is a Del Pezzo manifold.

9 at is be 100 By using the method of Main Theorem 1, we get a classification of polarized manifolds (X, ) with $n=\dim X\geq 3,$, and $\mathrm{b}\mathrm{s} L =\emptyset$ $g(l)=q(x)+2$. $([\mathrm{f}\mathrm{k}9])$ Main Theorem 2. Let (X, ) be a polarized manifold with $\dim X=n\geq 3$. Assume that Bs and $g(l)=q(x)+2$. $ L =\emptyset$ Then (X, ) is one of the following type: (1) $X$ is a double covering of $\mathrm{p}^{n}$ with branch locus being a smooth hypersurface of degree 6, and is the pull back of $\mathcal{o}_{\mathrm{p}^{n}}(1)f$ (2) (X, ) is a scroll over a smooth surface Y. Let $\mathcal{e}$ locally free sheaf of $\mathrm{y}$ rank two on such a that (X, ) $\cong$ $(\mathrm{p}_{s}(\mathcal{e}), H(\mathcal{E}))$. Then is either $(Y, \mathcal{e})$ $Y\cong \mathrm{p}_{\alpha}1\cross \mathrm{p}_{\beta}1$ (2-1) and, $\mathcal{e}\cong[h_{\alpha}+2h_{\beta}]\oplus[h_{\alpha}+h_{\beta}]$ where (resp. $H_{\beta}$) is the ample generator of $\mathrm{p}\mathrm{i}\mathrm{c}(\mathrm{p}_{\alpha})$ (resp. $H_{\alpha}$ $(\mathrm{p}_{\beta}))$ Pic. (2-2) $Y$ is the blowing up of $\mathrm{p}^{2}$ a point and, $\mathcal{e}\cong[2h-e]^{\oplus 2}$ where $H$ is the pull back of $\mathcal{o}_{\mathrm{p}^{2}}(1)$ and $E$ is the exceptional divisor, (2-3), $Y\cong \mathrm{p}(\mathcal{f})$ $\mathcal{f}$ where is a rank two vector bundle over an $-$ elliptic curve $C$ with $c_{1}(\mathcal{f})=1$ and, $\mathcal{e}=h(\mathcal{f})\otimes p^{*}(\mathcal{g})$ $\mathcal{g}$ $\mathrm{y}arrow C$ where $p$ : is the bundle projection and any rank two vector bundle on defined by a non splitting $C$ exact sequence $0arrow O_{C}arrow \mathcal{g}arrow O_{C}(X)arrow 0$, where $x\in C$. (3) There is a fibration : $f$ $Xarrow C$ over a smooth curve $C$ with $g(c)\leq 1$ such that every fiber of $F$ is a hyperquadric in $f$ $\mathrm{p}^{n}$ and $L_{F}=\mathcal{O}(1)$. Then $\mathcal{e}:=f_{*}(\mathcal{o}(l))$ is a locally free sheaf of rank $n+1$ on $C,$ $X\in 2H(\mathcal{E})+\pi^{*}(B) $ $\mathrm{p}(\mathcal{e})$ on for some line bundle $B$ $C_{f}$ on and, where $L=H(\mathcal{E}) _{X}$ $\pi$ is the projection, and is the tautological $\mathrm{p}(\mathcal{e})arrow C$ $H(\mathcal{E})$ $\mathrm{p}(\mathcal{e})$ line bundle on. We put $d=l^{n},$ and $e=c_{1}(\mathcal{e})_{f}$ $b=\deg B$. (3-1) If $g(c)=1$, then we have $n=3,$ $d=6,$ $e=4,$ $b=-2$, and $\mathcal{e}$ is ample.

10 \mathrm{a}}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i}$, 101 (3-2) If $g(c)=0$, then we have $3\leq d\leq 9_{f}e=d-3,$ $b=6-d$, and their lists are table 2 in [FI]. In the end, we propose a problem which is induced from Main Theorem 1. Problem 2. Classify $n$ -dimensional polarized manifolds with $g(l)=$ $q(x)+m$ and $h^{0}(l)\geq n+m$. If Bs $ L =\emptyset,$ $n\geq 3$, and $m\geq 0$, then we can get a classification of these polarized manifolds. We will report this in a future paper. REFERENCES [BLP] Beltrametti, M. C., Lanteri, A., and Palleschi, M., Algebraic surfaces containing an ample divisor of arithmetic genus two, Ark. Mat. 25 [BS] $\mathrm{b}\mathrm{e}\mathrm{l}\mathrm{t}_{\gamma [Fj 1] (1987), M. C. and Sommese, A. J., The adjunction theory of complex projective varieties, de Gruyter Expositions in Math. 16 (1995), Walter de Gruyter, Berlin, NewYork. Fujita, T., On polarized manifolds whose adjoint bundles are not semipositive, Advanced Studies in Pure Math. 10 (1985), [Fj2] Fujita, T., $Classifi_{Ca}ti_{on}$ of polarized $manifold_{s}$ of sectional genus two, [Fj3] the ProCeedings of Algebraic Geometry $\mathrm{c}\circ \mathrm{m}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{v}}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\gamma \mathrm{a}$ and in Honor of Masayoshi Nagata (1987), Fujita, T., contribution to $ c_{birat}ional$ geometry of algebraic varieties. Open problems, The 23rd Int. Symp. of the Division of Math. of the Taniguchi Foundation, Katata, August [Fj4] Fujita, T., $\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{d}_{0}\mathrm{n}$ Classification Theories of Polarized Varieties, Soc. Lecture Note Series, vol. 155, Cambridge University Press, [Fkl] Fukuma, Y., On polarized surfaces (X, L) with (L) $h^{0}$ $>$ 0, $\kappa(x)=$ 2, [Fk2] [Fk3] [Fk4] Math. and $g(l)=q(x)$, Trans. Amer. Math. Soc. 348 (1996), Fukuma, Y., A lower bound for the sectional genus of quasi-polarized surfaces, Geom. Dedicata 64 (1997), Fukuma, Y., A lower bound for sectional genus of quasi-polarized manifolds, J. Math. Soc. Japan 49 (1997), Fukuma, Y., On polarized 3-folds (X, L) with $g(l)=q(x)+1$ and $h^{0}$ (L), Ark. Mat. 35 (1997), $\geq 4$ [Fk5] Fukuma, Y., On polarized surfaces (X, L) with $h^{0}(l)>$ 0, $\kappa(x)\geq 0$ and $g(l)=q(x)+1$, Geom. Dedicata 69 (1998), [Fk6] Fukuma, Y., On the nonemptiness of the adjoint linear system of polarized manifolds, Can. Math. Bull. 41 (1998),

11 102 [Fk7] Fukuma, Y., On sectional genus of quasi-polarized 3-folds, Trans. Amer. Math. Soc. 351 (1999), [Fk8] Fukuma, Y., On quasi-polarized surfaces of general type whose sectional genus is equal to the irregularity, to appear in Geom. Dedicata. [Fk9] Fukuma, Y., On complex manifolds polarized by an ample line bundle of sectional genus $q(x)+2$, preprint (1998). [FI] Fukuma, Y. and Ishihara, H., Complex manifolds $p_{ola}\dot{n}zed$ by an ample and spanned line bundle of sectional genus three, Arch. Math. 71 (1998), [I] Ionescu, P., Generalized adjunction and applications, Math. Proc. Cambridge Philos. Soc. 99 (1986),